On right chain semigroups

Right chain semigroups are semigroups in which right ideals are linearly ordered by inclusion. Multiplicative semigroups of right chain rings, right cones, right invariant right holoids and right valuation semigroups are examples. The ideal theory of right chain semigroups is described in terms of prime and completely prime ideals, and a classification of prime segments is given, extending to these semigroups results on right cones proved by Brungs and Törner [H.H. Brungs, G. Törner, Ideal theory of right cones and associated rings, J. Algebra 210 (1998) 145-164].